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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Probability.InvariantIsing.Spectral.PositiveSpectralPartition
import OAI.Probability.InvariantIsing.Spectral.SpectralBallQuantizer
namespace OAI
/-! One-sided spectral rounding remains valid on the zero-mass cells. -/
noncomputable section
open MeasureTheory Set Metric
open scoped Topology Classical Function
namespace InvariantIsing
lemma positive_spectral_rounding_bounds {ι : Type*} [Fintype ι]
(μ : Measure ℝ) (S : ι → Set ℝ) (hcover : ∀ x, ∃ i, x∈S i)
(lam : ι → ℝ) (a b δ : ℝ)
(lo hi : {i // 0 < μ.real (S i)})
(hlo : lam lo-δ ≤ a) (hhi : b ≤ lam hi+δ)
(hcell : ∀ i, 0 < μ.real (S i) → ∀ x∈S i, dist (lam i) x ≤ δ)
(x : ℝ) (hx : x∈Icc a b) :
lam (positiveSpectralLabel (fun i => μ.real (S i)) lo
(spectralPartitionIndex S hcover x))-δ ≤ x ∧
x ≤ lam (positiveSpectralLabel (fun i => μ.real (S i)) hi
(spectralPartitionIndex S hcover x))+δ := by
let j := spectralPartitionIndex S hcover x
by_cases hp : 0 < μ.real (S j)
· have hc := hcell j hp x (spectralPartitionIndex_mem S hcover x)
rw [Real.dist_eq,abs_le] at hc
simp only [show spectralPartitionIndex S hcover x=j from rfl,
positiveSpectralLabel,dite_eq_left hp]
constructor <;> linarith
· simp only [show spectralPartitionIndex S hcover x=j from rfl,
positiveSpectralLabel,dite_eq_right hp]
exact ⟨hlo.trans hx.1,hx.2.trans hhi⟩
lemma positive_spectral_ball_cell_close {n : ℕ}
(μ : Measure ℝ) [IsProbabilityMeasure μ] (c r : Fin n → ℝ)
(a δ : ℝ) (hr : ∀ i, r i ≤ δ)
(hzero : μ (spectralBallCell (fun i => ball (c i) (r i)) none)=0)
(i : Option (Fin n)) (hi : 0 < μ.real (spectralBallCell (fun i => ball (c i) (r i)) i))
(x : ℝ) (hx : x∈spectralBallCell (fun i => ball (c i) (r i)) i) :
dist (i.elim a c) x ≤ δ := by
cases i with
| none =>
have hz : μ.real (spectralBallCell (fun i => ball (c i) (r i)) none)=0 :=
(measureReal_eq_zero_iff).mpr hzero
exact (not_lt_of_ge hz.le hi).elim
| some i =>
have hh : x∈ball (c i) (r i) := disjointed_subset (fun i => ball (c i) (r i)) i hx
simpa only [Option.elim_some,dist_comm] using (show dist x (c i) ≤ δ from hh.le.trans (hr i))
end InvariantIsing
end
end OAI
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